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Merton 1971 Optimum consumption and portfolio rules in a continuous-time model is an excellent application of the topic.

As @phi mentioned Arbitrage theory in Continuous Time by Bjork is an excellent resource as well.

Dixit and Pindyck Investment Under Uncertainty

The pitfall is essentially that in many problems we face the curse of dimensionality. This implies that PDE based approaches are less than practical (often practically impossible in many respects). There is one study which expands on Merton's 1971 work to FOUR ASSETS which essentially represented countries. Recent work by DeTemple Garcia and Rindisbacher 2003 solve the problem of intertemporal portfolio using Monte Carlo methods. See A Monte Carlo Method for Optimal Portfolios Parallel processing on existing infrastructure may make this a more practical approach to intertemporal asset allocation.